The metal used for the construction of thin curved shell structures such as ships, automobiles and aircraft is originally produced as thin, nominally flat sheet material. The formation of curved structures from flat sheet material generally involves some degree of elastic and/or plastic deformation of the material of the sheet referred to as strain. It is well known that there is a special class of surfaces, known as "developable" surfaces, which can be readily fabricated from sheet material, as they require only bending of the sheet rather than any degree of stretching, shrinking or other in-plane deformation. Consequently a developable surface is a highly preferred design for any application in which it will serve functionally. Such developable surfaces include, for example, cylindrical and conical surfaces.
However, developability is a highly restrictive property, and there are numerous situations in which requirements on the shape of the structure preclude or limit the use of developable surfaces. For example, hydrodynamic, aerodynamic, aesthetic or structural considerations may dictate the use of non-developable or "compound-curved" surfaces. In such cases the manufacturer must address the relatively difficult problem of forming these surfaces from flat sheet material.
A number of industrial "compounding" processes have been developed for this purpose. Most common is the use of matched dies, specialized tools whose high cost is distributed over a large number of mass-produced units. When the production volume is small, the known economical methods include spherical die pressing, peening, planishing and line heating. These methods utilize tools which are adaptable and inexpensive relative to matched dies. However, they demand relatively great amounts of skilled labor, because the forming processes are gradual and incremental and their control is highly empirical.
Two basic problems are present in the control of any of these incremental compounding processes. First, the machine operator knows only qualitatively, indirectly and empirically the complex relation between the application of the compounding tool and the resulting surface curvatures. Lacking quantitative guidance, the operator must depend on experience, training, judgment and intuition while incrementally approaching the desired curved configuration. This problem of achieving or implementing accurate and controlled compounding steps is referred to as Problem A.
Second, the in-plane strains which are a necessary part of the compounding process make it very difficult to relate the finished, 3-dimensional outline of a particular plate to the 2-dimensional shape of a flat starting "blank" to be cut from the original flat stock. Thus, excess material has to be left around the edges of a blank. The excess material at the edges is trimmed off in a trial-and-error fitting procedure referred to as "field trim" following compounding of the plate in order to fit the boundaries to adjacent plates or to other boundary lines on the 3-dimensional surface. In addition to requiring field trim steps, the excess material at the edges acts as a restraint to the compounding process. This problem of achieving or implementing accurate and controlled lofting of flat starting blanks for forming plates of compound curvature is referred to as Problem B.
The term "compound curvature", although commonly used in industry, does not appear either in the dictionary or in the literature of differential geometry, nor does the less common but synonymously used term "double curvature". However, the usage of both these terms is equivalent to the phrase "non-zero Gaussian curvature", which is founded in a rigorous geometric concept. "Compound curvature" is a qualitative term, not specifying the degree of curvature, while Gaussian curvature is quantitative.
A developable surface is characterized by having zero Gaussian curvature at every point. A plane is one example of a developable surface. Any developable surface such as a cylindrical surface or conical surface can be "developed" or flattened onto a plane by bending alone. A nondevelopable surface, one with non-zero Gaussian curvature, cannot.
The verb "to compound" is used herein to describe the introduction of Gaussian curvature into an originally plane piece of material by any mechanical or thermal process. A consequence of the geometric facts hereafter more fully developed is that compounding necessarily involves some degree of in-plane stretching or shrinking referred to as strain. Since the volume of material is conserved in each of these processes, the in-plane stretching or shrinking is generally accompanied by changes of thickness.
A brief description of several conventional mechanical compounding processes follows. These prior art methods are intended to be illustrative rather than all-inclusive.
Peening, (Planishing)
In this process repeated hammer blows are used directly to induce thickness reduction or thinning, thereby producing the required stretching in the plane of the material. The process is controlled by varying the number and/or weight of blows applied to various parts of the plate. For positive Gaussian curvature the thinning is concentrated in the center of the plate. For negative Gaussian curvature it is concentrated around the periphery.
Roller Planishing
In this process the thinning is accomplished by passing the work repeatedly between two rollers that are forced together. The process is controlled by varying the pressure of the rolls and the number of passes through the rolls made by each area of the plate.
Spherical Die Pressing
In this process a series of spherical matched dies of varying spherical radii are used to impress the appropriate amount of positive Gaussian curvature into each area of the plate. The dies are typically smaller in area than the plate, so that repeated pressings are required to form the complete plate. The die is changed if the curvature is not substantially constant. Saddle-shaped dies may be used similarly to impress negative curvature by die pressing.
Line Heating
This process consists of moving a point heat source over the plate in a pattern of lines that induce curvature by thermoplastic effects as set forth, for example, in "Line Heating", [1]; and Scully, K., [2]. The heat source can be adjusted so that its effect is predominantly bending, predominantly shrinking, or a combination of the two. The process is controlled by the intensity of heat, the speed of movement over the plate, and the pattern of lines where heat is applied. When properly controlled, the effect of one shrinking heat line is to draw the plate together by a predictable increment of distance perpendicular to the line. To produce positive Gaussian curvature, heat lines are concentrated toward the periphery of the plate. To produce negative Gaussian curvature the heat lines are concentrated toward the center.
The unifying feature of all these incremental compounding methods is that Gaussian curvature is produced by in-plane strain. The patterns of strain vary according to the process. In peening, planishing and die pressing the process possesses a radial symmetry that implies an isotropic strain pattern. Each hammer blow forces material outward equally in all directions. The sum of all the incremental hammer blows spread over an area of the work is therefore an isotropic expansion in the plane of the work. A small circle marked on the blank ends up as a somewhat larger circle on the finished plate, having been expanded equally in all directions.
Roller planishing contains a preferred direction along the axis of the rollers, and therefore produces an orthotropic strain pattern. A small circle marked on the blank is enlarged more along one axis than the other, producing an ellipse. In line heating the fundamental strain pattern is yet more pronouncedly anisotropic. The contraction produced by a heat line is entirely in the direction perpendicular to the line.
The various compounding processes share Problem A to varying degrees. That is, the operator of the compounding machine or tool knows only empirically, qualitatively, and by trial and error process how to apply the tool in order to achieve a desired configuration of compound or non-zero Gaussian curvature. The most direct shape control is available in die pressing. If the operator is told the specified Gaussian curvature for each area of the plate, he can select the dies accordingly to be applied to the blank. (Compensation for springback, or elastic strain of the plate accompanying release of pressure may have to be allowed empirically in the die selection.) In each of the other methods, the relationship between the application of the process, e.g. the intensity of peening in various areas of the plate, and the resulting curvature, is fairly obscure. Workers develop a sense of this relationship through training and practice, but the application of this experience is an art rather than a science.
Problem B is shared equally by all compounding processes. That is, there is no accurate method for lofting blanks to be compounded into plates with zero or minimal field trim. On a non-developable plate, the in-plane strains required to produce the requisite Gaussian curvature also produce distortions of any in-plane measurements intended to relate the outline of the blank to the finished outline of the plate. The nature of these distortions is sufficiently complex to have defeated all attempts to loft accurate blanks from 3-dimensional specifications of the patch shape and location. Various approximate, empirical methods are in use, for example as described by Newton, R. N., [3]; and Grilliat, J. (editor), [4]; but the solutions they provide are known to be more or less inaccurate and unreliable.